At a very basic level, Bitcoin is just a digital file or ledger that contains names and balances, and people exchange money by changing this file. When Bob sells Carol a lawn mower for 5.2 Bitcoins, Bob’s balance goes up by 5.2, and Carol’s down by 5.2. There’s no gold or government issued money backing these numbers. Bob is prepared to part with his lawnmower because he believes someone will add 5.2 Bitcoins to his account in the ledger. Quite simple really.
The beauty of the system is it is based on belief in the system and not on trust between parties in the system. Bob does not even have to know who Alice is.
So who maintains this ledger and makes sure no one cheats? Bitcoin aims to avoid any centralized control, so every participant maintains their own copy of the ledger. Simply stated – every transaction is published and everyone updates their copy of the ledger. It gets a little more complicated than that and some issues have to be designed for
How to prove who is the owner of an account – some form of digital signature is needed (listen out for public keys and private keys and digital wallets).
How to check a signature was actually created by the message sender without sharing both the private and public key (listen out for cryptographic hash)
How to know when a message was sent and in what order to process them (network delays and the decentralised nature makes this a challenging problem). Here the video starts to talk about chains and linking.
How to bring a ledger up to date when it has been off line for a while
Solving these problem is commonly called “mining,” as this is how new Bitcoin enters the system, but the main purpose of the math is to make sure everyone’s ledgers agree.